PSFC/JA-01-35
EMISSION OF ELECTRON BERNSTEIN WAVES
A. K. Ram, A. Bers, G. Taylor* and P. Efthimion*
December 2001
Plasma Science & Fusion Center
Massachusetts Institute of Technology
Cambridge, Massachusetts 02139, U.S.A.
* Princeton Plasma Physics Laboratory
Princeton, NJ 08543, U.S.A.
This work was supported by the U.S. Department of Energy Contracts No. DE-FG02-91ER-54109 and DE-FG02-99ER-54521. Reproduction, translation, publication, use and disposal, in whole or part,
by or for the United States Government is permitted.
To appear in Proceedings of the 14th Topical Conference on Radio
Frequency Power in Plasmas, Oxnard, California, May 7–9, 2001.
i
ii
Emission of Electron Bernstein Waves1
A. K. Ram, A. Bers, G. Taylor,∗ and P. Efthimion∗
Plasma Science and Fusion Center, M.I.T., Cambridge, MA 02139, U.S.A.
∗
Princeton Plasma Physics Laboratory, Princeton, NJ 08543. U.S.A.
INTRODUCTION
We have previously shown that the high-β NSTX-type plasmas can be heated
by waves in the electron cyclotron range of frequencies by mode converting the
X mode or the O mode to electron Bernstein waves (EBW) at the upper hybrid
resonance [1]. From ray tracing analysis we have also shown that EBWs are locally
and strongly absorbed at the Doppler shifted electron cyclotron resonance or its
harmonics. For spherical tokamak plasmas, the location of the EBW energy deposition is significantly different for excitations on or away from the equatorial plane
[1,2]. The strong and localized absorption implies that thermal emission of EBWs
can occur for frequencies corresponding to the local electron cyclotron frequency.
We have developed a kinetic model for mode conversion of the X mode and the
O mode to EBWs. Using this model we evaluate, analytically and numerically,
properties of the X mode and the O mode that would be observed in the vacuum
region from mode conversion of EBWs at the upper hybrid resonance (UHR). The
mode conversion studies have shown that the optimum regime, in frequency and
parallel wavelength space, for conversion of the X mode to EBWs is different from
that of the O mode conversion to EBWs. In particular, X-mode couples efficiently
to EBWs for longer parallel wavelengths and smaller frequencies than the O-mode.
Our results show that the emission of EBWs provide a good indication of the possible scenarios for conversion to EBWs for heating and driving currents in overdense
plasmas.
The excitation and emission of EBWs has been studied on the Wendelstein
7-AS [3]. More recently, experiments on CDX-U and NSTX [4] and MST [5]
have observed emission of EBWs via the mode conversion process and studied its
dependence on the edge properties of the plasma.
In this paper we formulate the kinetic description of the mode conversion process
which includes the propagating EBWs. From numerical solutions of this model we
find that the conversion efficiency of EBWs to the X and O modes is the same as
the conversion efficiency of the X mode to EBWs and of the O mode to EBWs.
Consequently, one can design an EBW excitation experiment based completely on
the observed characteristics of the emitted X and O modes.
1)
Work supported by DoE Grants DE-FG02-91ER-54109 and DE-FG02-99ER-54521.
APPROXIMATE KINETIC DESCRIPTION OF THE
MODE CONVERSION
Our previous studies have shown that efficient coupling between EBWs and the
X or O modes occurs when the UHR is in the steep density gradient regions near
the edge of the plasma. For this region we use the slab geometry model where the
x axis is the direction of inhomogeneity and corresponds to the radial direction, the
y axis is along the poloidal direction, and the z axis is along the toroidal direction.
The equilibrium magnetic field is assumed to be sheared with the form
0 (x) ≡ By (x)ŷ + Bz (x)ẑ = B0 (x) sin Ψ(x)ŷ + B0 (x) cos Ψ(x)ẑ
B
(1)
0 and the z-axis. The variation of the fields in the y
where Ψ is the angle between B
and z directions is assumed to be of the form exp(iky y + ikz z) where ky and kz are
the appropriate components of the wave vectors. The time dependence is assumed
to be of the form exp(iωt) where ω is the wave angular frequency. Assuming that
the EBWs are electrostatic, we find that the approximate full wave description of
the propagation of the X mode, O mode, and the EBWs is given by [1]
↔
dF
(2)
= i AK · F
dξ
where ξ = ωx/c is the normalized spatial variable,
F T = [Ex Ey Ez (iχ̃E ) cBz (−cBy )]
(3)
x
is the transpose of the field vector F , Ex = (dEx /dξ),

0
0
0
−χ−1
1
 ny
0
0
0

 n
↔
0
0
0

z
=
A 
χxy
χxz
 Kxx
0

 −χxy
Kyy − n2z χyz + ny nz
0
0
−χxz χyz + ny nz Kzz − n2y
↔

↔
↔
K=I + χ
−ωp2  1
χ=
 iωcz
ω 2 − ωc2
−iωcy
↔
0
1
0
ny
0
0
−iωcz
2
1 − ωcy
−ωcy ωcz

0
0 

1 


nz 

0 
0
(4)
(5)

iωcy
−ωcy ωcz 

2
1 − ωcz
(6)
where ωp (x) is the electron plasma angular frequency, ωcy (x) = eBy (x)/me
ωcz (x) = eBz (x)/me are the electron cyclotron
angular frequencies for the poloidal
2
2 ,
and toroidal fields, respectively, ωc = ωcy + ωcz
χ1 =
vT
ωc
2 ωp2
ω2
−
ω 2 − ωc2 ω 2 − 4ωc
(7)
and vT = T (x)/m is the electron thermal velocity for corresponding to the temperature T (x).
EXCITATION AND EMISSION OF ELECTRON
BERNSTEIN WAVES
For NSTX-type parameters discussed in [1], and for a wave frequency of 14 GHz,
Fig. 1a shows the mode conversion efficiency CXB for the excitation of EBWs
when a X mode is launched from outside the plasma. Figure 1b shows the mode
conversion efficiency COB when, for the same parameters when an O mode is
launched from the outside.
fractional power
1
1
CXB
R
RX
O
0.5
0.5
R
R
O
0
0
0.2 0.4 0.6
nz = ckz/ω
X
0.8
0
0
COB
0.2 0.4 0.6
n = ck /ω
z
(a)
(b)
0.8
z
FIGURE 1. The fraction of the input power, as a function of kz for ky = 0, that is reflected
out on the X mode RX , on the O mode RO , and mode converted to Bernstein waves: (a) for
incoming X mode the conversion coefficient is CXB ; (b) for incoming O mode the conversion
coefficient is COB .
When EBWs are emitted from inside the plasma, they propagate out to the
edge where they can couple to the X and O modes near the UHR. Some of the
EBW power reaching the UHR will get reflected back into the plasma. For the
same parameters as discussed above, Fig. 2 shows the fraction of the emitted EBW
power, as a function of kz , that is converted to the X mode EX , to the O mode EO ,
and reflected back into the plasma RB . Upon comparing the results from Figs. 1a
and 1b with those given in Fig. 2 we note the following relationships
EX = CXB
and
EO = COB
(8)
Thus, the fraction of the EBW emitted power that is mode converted to the X
mode and the O mode is the same as the power that is mode converted from the X
mode to the EBW and, separately, from the O mode to EBW. Since EX + EO ≤ 1,
it follows that if the plasma conditions are such that the X-B conversion process is
optimized then the O-B conversion process is not effective, and vice-versa. We have
1
fractional power
0.8
EX
R
B
0.6
0.4
E
O
0.2
0
0
0.2
0.4
nz
0.6
0.8
FIGURE 2. The fraction of the outgoing EBW power, as a function of kz for ky = 0, that is
converted to the X mode EX and on the O mode EO . RB is the fraction of the emitted EBW
power that is reflected back into the plasma at the upper hybrid resonance.
noted this effect before when studying the mode conversion excitation of EBWs
[1]. It was found that the X-B and the O-B conversion processes were optimized
in, essentially, mutually exclusive region of the parameter space spanned spanned
0 ). Consequently, from the
by (ω, k) (where k is the component of k parallel to B
relationship in (8) we can conclude that the experimental design for the excitation
of EBWs can be completely based on the EBW emission characteristics of the
plasma.
REFERENCES
1. Ram, A. K., and Schultz, S. D., Phys. Plasmas 7, 4084 (2000).
2. Forest, C. B.,et al., Phys. Plasmas 7, 1352 (2000).
3. Laqua, H. P., et al., Phys. Rev. Lett. 78, 3467 (1997); Laqua, H. P., et al., Phys. Rev.
Lett. 81, 2060 (1998);
4. Taylor, G., et al., in Proceedings of this Conference.
5. Chattopadhyay, P. K., et al., in Proceedings of this Conference.